Advertisement

Supersimple Theories

  • Frank O. Wagner
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 503)

Abstract

Recall from Definition 2.8.12 that a simple theory is supersimple if for all finite tuples ā and all A there is a finite subset A 0A with \( { \downarrow _{A0}}A. \) The importance of supersimplicity stems from the fact that it allows a global, ordinal-valued rank, invariant under definable bijections, which orders definable sets and types and is compatible with independence. In fact, there are two (main) ranks; one suitable for complete types and one suitable for partial types. In this chapter, we shall again assume that we work in a simple theory (which need not be supersimple).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliographical Remarks

  1. [143]
    Saharon Shelah. Stability, the f.c.p., and superstability; model-theoretic properties of formulas in first-order theories. Annals of Mathematical Logic, 3: 271–362, 1971.CrossRefzbMATHGoogle Scholar
  2. [147]
    Saharon Shelah. Classification Theory. North-Holland, Amsterdam, The Netherlands, 1978.Google Scholar
  3. [78]
    Daniel Lascar. Types définissables dans les théories stables. Comptes Rendus de l’Académie des Sciences à Paris, 276: 1253–1256, 1973.zbMATHMathSciNetGoogle Scholar
  4. [81]
    Daniel Lascar. Ranks and definability in superstable theories. Israel Journal of Mathematics, 23: 53–87, 1976.CrossRefzbMATHMathSciNetGoogle Scholar
  5. [116]
    Bruno P. Poizat. Déviation des Types. PhD thesis, Université Pierre et Marie Curie, Paris, France, 1977.Google Scholar
  6. [70]
    Byunghan Kim and Anand Pillay. Simple theories. Annals of Pure and Applied Logic, 88: 149–164, 1997.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [147]
    Saharon Shelah. Classification Theory. North-Holland, Amsterdam, The Netherlands, 1978.Google Scholar
  8. [83]
    Daniel Lascar. Ordre de Rudin-Keisler et poids dans les théories superstables. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 28: 411–430, 1982.CrossRefMathSciNetGoogle Scholar
  9. [84]
    Daniel Lascar. Relations entre le rang U et le poids. Fundamenta Mathematicae, 121: 117–123, 1984.zbMATHMathSciNetGoogle Scholar
  10. [60]
    Hyttinen. Remarks on structure theorems for wi -saturated models. Notre Dame Journal of Formal Logic, 36: 269–278, 1995.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [106]
    Anand Pillay. Geometric Stability Theory. Oxford University Press, Oxford, UK, 1996.zbMATHGoogle Scholar
  12. [125]
    Bruno P. Poizat. Cours de théorie des modèles. Nur Al-Mantiq WalMa’rifah, Villeurbanne, France, 1985.Google Scholar
  13. [17]
    Steven Buechler. Lascar strong type in some simple theories. Journal of Symbolic Logic, to appear.Google Scholar
  14. [19]
    Steven Buechler, Anand Pillay, and Frank O. Wagner. Elimination of hyperimaginaries for supersimple theories. Preprint, 1998.Google Scholar
  15. [68]
    Byunghan Kim. Simplicity, and stability in there. Preprint, 1999.Google Scholar
  16. [111]
    Anand Pillay. Normal formulas in supersimple theories. Preprint, 1999.Google Scholar
  17. [160]
    Frank O. Wagner. Groups in simple theories. Preprint, 1997.Google Scholar
  18. [163]
    Frank O. Wagner. Hyperdefinable groups in simple theories. Preprint, 1999.Google Scholar
  19. [46]
    Ehud Hrushovski. Contributions to Stable Model Theory. PhD thesis, University of California at Berkeley, Berkeley, USA, 1986.Google Scholar
  20. [167]
    Boris Zil’ber. Groups and rings whose theory is categorical (in Russian). Fundamenta Mathematicae, 95: 173–188, 1977.zbMATHMathSciNetGoogle Scholar
  21. [14]
    Chantal Berline and Daniel Lascar. Superstable groups. Annals of Pure and Applied Logic, 30: 1–43, 1986.CrossRefzbMATHMathSciNetGoogle Scholar
  22. [25]
    Zoe Chatzidakis and Ehud Hrushovski. The model theory of difference fields. Transactions of the American Mathematical Society, 351 (8): 2997–3071, 1999.CrossRefzbMATHMathSciNetGoogle Scholar
  23. [126]
    Bruno P. Poizat. Groupes Stables. Nur Al-Mantiq Wal-Ma’rifah, Villeurbanne, France, 1987.Google Scholar
  24. [133]
    Joachim Reineke. Minimale Gruppen. Zeitschrift für Mathematische Logik, 21: 357–359, 1975.zbMATHMathSciNetGoogle Scholar
  25. [93]
    Angus Macintyre. On wi-categorical theories of fields. Fundamenta Mathematicae, 71: 1–25, 1971.zbMATHMathSciNetGoogle Scholar
  26. [31]
    Gregory Cherlin and Saharon Shelah. Superstable fields and groups. Annals of Mathematical Logic, 18: 227–270, 1980.CrossRefMathSciNetGoogle Scholar
  27. [113]
    Anand Pillay and Bruno P. Poizat. Corps et chirurgie. Journal of Symbolic Logic, 60: 528–533, 1995.CrossRefzbMATHMathSciNetGoogle Scholar
  28. [49]
    Ehud Hrushovski. Pseudo-finite fields and related structures. Preprint, 1991.Google Scholar
  29. [27]
    Zoe Chatzidakis and Anand Pillay. Generic structures and simple theories. Annals of Pure and Applied Logic, 95: 71–92, 1998.CrossRefzbMATHMathSciNetGoogle Scholar
  30. [114]
    Anand Pillay, Thomas Scanlon, and Frank O. Wagner. Supersimple division rings. Mathematical Research Letters, pages 473–483, 1998.Google Scholar
  31. [61]
    Nathan Jacobson. Basic Algebra II (second edition). Freeman, New York, USA, 1989.Google Scholar
  32. [138]
    Jean-Pierre Serre. Local Fields. Springer-Verlag, Berlin, Germany, 1979.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Frank O. Wagner
    • 1
  1. 1.Institut Girard DesarguesUniversité Claude Bernard (Lyon-1)VilleurbanneFrance

Personalised recommendations